Optimal. Leaf size=322 \[ -\frac{5 (9 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.521904, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{5 (9 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4} b^{3/4}}+\frac{5 (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} b^{3/4}}-\frac{5 (9 A b-a B)}{16 a^3 b \sqrt{x}}+\frac{9 A b-a B}{16 a^2 b \sqrt{x} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b \sqrt{x} \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 90.2279, size = 298, normalized size = 0.93 \[ \frac{A b - B a}{4 a b \sqrt{x} \left (a + b x^{2}\right )^{2}} + \frac{9 A b - B a}{16 a^{2} b \sqrt{x} \left (a + b x^{2}\right )} - \frac{5 \left (9 A b - B a\right )}{16 a^{3} b \sqrt{x}} - \frac{5 \sqrt{2} \left (9 A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{13}{4}} b^{\frac{3}{4}}} + \frac{5 \sqrt{2} \left (9 A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{13}{4}} b^{\frac{3}{4}}} + \frac{5 \sqrt{2} \left (9 A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}} b^{\frac{3}{4}}} - \frac{5 \sqrt{2} \left (9 A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}} b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**(3/2)/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.495145, size = 285, normalized size = 0.89 \[ \frac{\frac{32 a^{5/4} x^{3/2} (a B-A b)}{\left (a+b x^2\right )^2}+\frac{5 \sqrt{2} (a B-9 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}+\frac{5 \sqrt{2} (9 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}+\frac{10 \sqrt{2} (9 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{3/4}}-\frac{10 \sqrt{2} (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{3/4}}+\frac{8 \sqrt [4]{a} x^{3/2} (5 a B-13 A b)}{a+b x^2}-\frac{256 \sqrt [4]{a} A}{\sqrt{x}}}{128 a^{13/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^3),x]
[Out]
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Maple [A] time = 0.026, size = 363, normalized size = 1.1 \[ -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{13\,{b}^{2}A}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{5\,Bb}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{17\,Ab}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{9\,B}{16\,a \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{45\,\sqrt{2}A}{128\,{a}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}A}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}A}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{128\,{a}^{2}b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}B}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^(3/2)/(b*x^2+a)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258484, size = 1177, normalized size = 3.66 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**(3/2)/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.272413, size = 405, normalized size = 1.26 \[ -\frac{2 \, A}{a^{3} \sqrt{x}} + \frac{5 \, B a b x^{\frac{7}{2}} - 13 \, A b^{2} x^{\frac{7}{2}} + 9 \, B a^{2} x^{\frac{3}{2}} - 17 \, A a b x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b^{3}} - \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{3}} + \frac{5 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(3/2)),x, algorithm="giac")
[Out]